Research Into Building Vibrations
by
Diane Lee Bosuego Floresca
B.S. Civil and Environmental Engineering (2001)
University of California, Berkeley
Submitted to the Department of Civil and Environmental Engineering
in Partial Fulfillment of the Requirements for the Degree of
Master of Engineering in Civil and Environmental Engineering
at the
Massachusetts Institute of Technology
June 2003
© 2003 Massachusetts Institute of Technology
All rights reserved
Signature of Author...
Department of Civil and Environmental Engineering
May 9, 2003
0
Certified by....
J
'7
-I
Accepted by..........
. .
K
........................
Jerome J. Connor
Professor of Civil Engineering
Thesis Supervisor
................. I*.........................
Oral Buyukozturk
Professor of Civil Engineering
ental
Committee
on Graduate Studies
Chairman, Departm
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
JUB 0 2 2003
LIBRARIES
Acknowledgements
First,I would like to thank my family for their continuing support throughout my life.
Without them, I could not have come this far.
Next, I would like to thank all of the Professors and Lecturers here at MITfrom whom I
had the great honor of learning,especially Jerome Connor and John Macomber.
Lastly, I would like to thank Paul Kassabianand Lisa Grebnerfor all their help and
guidance during the school year.
This is dedicated to Jo Ann B. Floresca
May 7, 2003
2
RESEARCH INTO BUILDING VIBRATIONS
by
DIANE LEE BOSUEGO FLORESCA
Submitted to the Department of Civil and Environmental Engineering
on May 7, 2003 in partial fulfillment of the
requirements for the Degree of Master of Engineering in
Civil and Environmental Engineering
ABSTRACT
Underground and surface arteries for vehicle or railway traffic can create vibrations that
travel to nearby buildings. These vibrations can cause structural damage or human
discomfort. Displacement time histories collected from buildings abutting the central
surface artery were used to drive mathematical models so that asphaltic and polymeric
bearings could be studied as possible passive mitigators of such vibrations. Neither
material attenuated vibrations to below threshold levels for human annoyance, but they
could dampen levels to resist structural damage if enough material was used to bring the
apparent natural frequency away from the range characteristic of traffic vibration. In
addition, for resonant cases, the materials did not create enough damping force to counter
the inertia of heavy structures, because the materials were too stiff and the displacements
and velocities too small. For new construction, it is suggested that these vibrations
should be prevented from entering the foundation area by surrounding the foundation
with a concrete wall or absorbent foam blocks. For retrofits, polymeric or asphaltic pads
could be used and would be relatively easy to install.
Thesis Supervisor: Jerome J. Connor
Title: Professor of Civil Engineering
3
Table of Contents
1.0 Introduction
5
2.0 Vibration
2.1 Traffic Vibrations 8
2.2 Characteristic Wave Motion 9
2.3 Microtremors 10
2.4 Parameter to Describe Vibrations
6
11
3.0 Threshold Levels
3.1 Human Annoyance 12
3.2 Damage to Buildings 12
14
3.3 Damage Criteria
12
4.0 Theory and Methodology
4.1 Damping
18
4.2 Viscoelastic Theory 19
18
5.0 Materials
22
5.1 Asphalt
5.2 Viscoelastic Polymers
22
24
6.0 Mathematical Model
6.1 Single Degree of Freedom System 26
29
6.2 Damping Device
6.3 Preliminary Analysis: Pure Sinusoidal Excitation
Model Parameters 32
Controlling Velocity 34
Input Vibration Data 37
Model and Test Procedure
8.0 Results
8.1 Hysteresis Loops
31
32
7.0 Simulation Procedure
7.1
7.2
7.3
7.4
26
39
41
44
9.0 Feasibility
46
9.1 Vertical Load Capacity
9.2 Possible Design of Isolation System47
49
9.3 Other Materials: Foam
46
Conclusion
51
Appendix A: Results from Simulation
52
Appendix B: Matlab script used to run simulation
55
4
1.0 Introduction
With the successful completion of Boston's subsurface Central Artery/Tunnel (CA/T)
Project, many other cities around the United States may want to begin similar projects to
help alleviate traffic congestion. Underground and surface arteries for vehicle or railway
traffic can create vibrations that travel to nearby buildings. These vibrations can cause
structural damage or human discomfort. Studies have shown that humans are annoyed by
vibration levels right at the threshold to where they can just perceive them, especially if
the source is unknown. Such vibrations can disrupt sleep and the work environment.
In addition, these vibrations are undesirable for high performance chip manufacturing
plants or genetic laboratories, whose vibration tolerances are significantly more stringent
than standard purpose commercial buildings. For these types of structures, the building
itself accounts for approximately 25 percent of the total facility cost, and the building is
constructed to remain "state of the art" for up to 10 years. Because fabrication plants
produce products at microscopic dimensions, the slightest vibration could shift chipmaking equipment calibrations.
For these reasons, it is necessary to study the ways in which these types of vibrations can
be effectively, easily, and cheaply mitigated.
5
2.0 Vibration
Vibration is an oscillatory movement around a state of equilibrium. Vibration can be
described by particle position (or displacement), particle velocity (change in
displacement over time), or particle acceleration (change in velocity over time).
There are five main types of deterministic vibrations. Deterministic means that the
vibration motion can be modeled with some mathematical function, in order to predict
with some probability the characteristics of the vibration.
Deterministic
Non Periodic
Periodic
Sinusoidal
Complex
Periodic
Quasi
Periodic
Transient
Figure 2.1 Organization of Types of Vibrations
6
77me
vibr#diOfl
a)
b)
Composed
Tdra~on
(demnistic)
Sftaiionary
)
9 ejmewy
e)
Ld
v'i$-
O
Figure 2.2 Illustrations of Different Types of Vibrations (Holmberg 1984)
7
2.1 Traffic Vibrations
When a vehicle rolls on a road, vibrations are created due to depressions made by the
weight of the vehicle, with the pressure rapidly changing as the wheel rolls. The energy
is transmitted from the vehicle to the ground continuously. If the roadway is bumpy,
shocks are induced and the energy propagates as waves to the environment. These waves
are mostly Rayleigh-type surface waves, but there are also vertically polarized shear
waves. The Figure below shows the various parameters by which the particle velocity
may be affected.
A
Speed
A
r*4
0.u-
4ey
too0 ..
*/q A^
.UIt
ffmRSi_Ave-
pfle
100
M
piae
Figure 2.3 Parameters Affection Traffic Vibration Level (Holmberg 1984)
8
Because the propagation of the surface waves takes place in horizontal ground layers,
these waves are more slowly attenuated than the volume waves. Changes in depth and
composition of the underground also have large effects on the surface waves. As heavy
vehicles are often long and all the wheels function as vibration sources, the vibrations
have a long period. They also have rather low frequencies and can therefore cause
vibrations of both buildings and separate building elements.
Measurements taken in Alvangen, Sweden found that the frequency of vibrations from
railroad traffic nearby varied around 4 Hz. Grootenhuis sites studies indicating wheelrail interaction from underground railway traffic causing vibrations to nearby buildings to
be on the order of 15-30 Hz, depending on the speed of the train.
2.2 Characteristic Wave Motion
Most significant in our discussion of microtremors due to traffic vibration is the
Rayleigh-wave (R-wave). The wave particles describe a retrograde elliptical motion,
where the vertical component has its maximum just below surface, but diminishes
thereafter relatively rapidly with the depth. R-waves have the lowest velocity, compared
to shock waves (P-waves) and shear waves (S-waves). The wave velocity is determined
by the elastic properties of the material.
9
2.3 Microtremors
Vibrations induced by vehicular and railway traffic can be characterized as microtremors,
which are low amplitude, low frequency ground motions, and are caused by a number of
other sources including earthquake aftershocks and construction. It is important to study
microtremors because they can cause damage to nearby structures and bring annoyance
and discomfort to humans.
Noise vibrations, like microtremors, are said to be random, although governed by natural
law. Because of this, they are unlike deterministic vibrations because they cannot be
described in mathematical expressions but rather with statistical means, which in
principle make it very difficult to predict their behavior.
The most common terms used to describe noise vibrations are r.m.s. (root mean square)
(standard deviation), power density spectrum, autocorrelation function, and probability
density function. Traffic noise is non stationary, in that these parameters are not
independent of absolute time (during a long enough, but limited time space) but rather
vary with time and have average values. In many cases, however, the vibration can
approximately be regarded as being deterministic, for the sake of design.
10
2.4 Parameter to Describe Vibrations
In describing vibrations, the displacement, velocity, or acceleration time histories can be
characterized by several important parameters.
" Peak Value: X, is the maximum absolute value of the vibration during a time
interval.
" R.M.S.-value: rms is defined as
t2
Jx2dt
rMs
tl
t 2 -ti
=
For a pure sine wave
A
rms =
x
* The mean value is often defined to be 0.
*
Standard Deviation:
t2
J(x-x)2dt
ti
t2
-t
If the mean value is zero, then the standard deviation equals the rms.
11
3.0 Threshold Levels
3.1 Human Annoyance
Man interprets vibrations as warning signals, although it is generally obvious as a soon as
the source is identified that no risk exists. The long-term effects from disturbances
depend as much on the acceptance of the source as the threshold value of the disturbance.
It is widely accepted that threshold levels for human annoyance are right above the level
at which they are possible to perceive. In the ISO 2631/DAD 1 (1980) standard, values
of 0.14 and 0.2 mm/s are indicated for frequencies exceeding 8 Hz for night and daytime,
respectively. The threshold for perception is at about 0.1 mm/s for corresponding
frequencies.
3.2 Damage to Buildings
Damage due to vibrations depends on so many factors, that it is very difficult to assess
the risk. The method of construction of the building, the vibration's character (intensity,
frequency range, wave length), static stress of the building, and how much the original
12
values have been increased by settlements, dampness and temperature variations all
contribute to the damage of a building.
There are few generally accepted standard values resulting from the fact that the number
of well-documented damage cases is very limited. Vibrations from traffic seldom reach
levels where they can cause direct vibration damage. Generally, accelerated aging
results. A reasonable starting point then is to relate the intensity of stress caused by the
vibrations from an external source to the inevitable stresses that a building withstands due
to the indoors environment and climatological factors.
13
3.3 Damage Criteria
The table below shows a compilation of recommended limit values used when judging
the risk for damage by ground vibrations in normal residential areas. Normal residential
areas indicate houses with foundations and joists of concrete, outer walls of brick and
intermediate partitions of plastered, compact light concrete.
Sand,
Gravel,Clay
Moraine, Slate
Stone, Soft limestone
Granite gneiss,
firm limestone,
quartzite
sandstone,
Results in
normal
residential area
diabase
Vertical particle
velocity v
[mm/s]
18
35
70
No noticeable
cracking
30
55
110
Fine cracks, and
fall of plaster
(threshold value)
40
80
160
Cracking
60
115
230
Serious Cracking
Table 3.1 Limit Values for Damage in Buildings. Langefors and Kihlstrom (1967).
Below is the graphical representation of the limiting values shown in the table above. For
frequencies exceeding 40 Hz, the particle velocity is the criterion, but at lower
frequencies the displacement represents the criterion.
14
PaMice
too
5D
200
to
10z
10
b
0
oo
10
Figure 3.1 Illustration of Threshold Values (Holmberg 1984)
Criteria For Damage and Recommendations
1.) Direct damage from vibrations on buildings at blasting
2.) Recommended upper limit for blasting
3.) Recommended upper limit for piling, sheet piling, vibratory compactors, dynamic
deep compaction and traffic
4.) Disturbing vibrations to human beings.
15
Studer and Suesstrunk (1981) have compiled the results of vibration measurements in
Switzerland, carried out later than 1960. There are for nominated criteria for four classes
of structures and the associated standards for vibrations caused by various construction
procedures and traffic.
Building Class
I. Industrial buildings
Frequency range
where the standard
value is applicable
10 - 30
Maximum resultant
vr [mn/s] of the
particle velocities
12
Estimated maximum
vertical particle
velocity v-z [mm/s]
7.2 - 12
of reinforced
concrete, steel
constructions
30 - 60
12 - 18
II. Buildings on
10 - 30
8
7.2
18
-
4.8 - 8
concrete foundation.
Concrete walls or
30 - 60
bricked walls
III. Buildings with
bricked cellar walls.
10 - 30
Upper apartment
30 - 60
floors on wooden
beams
IV. Especially
sensitive buildings
and historical
buildings
8
12
-
-
12
-
3.0 - 5
5
5
4.8
8
10-30
3
30- 60
3-5
3.0
-
8
1.8-3
1.8 -5
Table 3.2 Threshold Levels of Building Damage
16
Bonde, Rundqvist and others (1981) propose limiting values for vibrations caused by
vehicles (traffic). The buildings are assumed to be founded on clay or loosely layered
sand. The recommendations refer to the case when the number of vehicles with a total
weight of 10 tons exceeds 50 vehicles per 24 hours. If the number is lower, the limiting
values can be increased by 50%. A summary of the results are shown below.
Type of building and foundation
Recommended limit value [mm/s]
1
Especially sensitive buildings and buildings of
cultural and historical value
Newly-built buildings and/or foundations on a
foot plate
Buildings on cohesion piles
2
Buildings on bearing piles or friction piles
5
3
Table 3.3 Threshold Levels for Traffic Vibrations
17
4.0 Theory and Methodology
4.1 Damping
Damping refers to the dissipation of vibrational energy. All physical systems have some
inherent damping, but the level of damping can be augmented to increase energy
dissipation in particular vibration modes. In this way, the response of a structure driven at
a resonant frequency can be greatly decreased. This in turn can significantly reduce
overall motion or acceleration of the structure. The damping process dissipates or
absorbs the energy input from external excitations by transferring it to other mechanical
forms such as heat, sound, or strain energy.
Viscoelastic damping is a form of passive damping where the input energy is dissipated
by a transfer to heat and strain energy to the viscoelastic material. Since for viscous
materials the damping force is a function of the rate of deformation, these forces are out
of phase with the elastic forces in the system and do not add to the total force at the
maximum displacement.
18
4.2 Viscoelastic Theory
In simple elastic mechanics, the relationship between stress and strain is linear and based
on Young's Modulus for simple uniaxial strain.
- =E
(4.1)
And for simple shear strain, the relationship is:
r = G,7 , where Ge is the Elastic Shear Modulus.
(4.2)
This shows that since the ratio of stress to strain is constant, all strain energy created will
be stored as potential energy and released when the elastic material decompresses. In
reality, there are no materials that are 100% elastic.
Viscoelastic materials behave such that the rate of energy dissipation becomes
significant, and the modulus can be adjusted to take into account the energy dissipative
nature of the material.
E*= Ei(1+ 7i)
(4.3)
where Elis the storage modulus and ilis the loss factor, creating an imaginary term.
For 100% viscous materials, the relationship is:
19
o- =
(4.4)
(
Ee
where the strain can be modeled as a sinusoidal wave form, resulting in:
(4.5)
E = . sin(Qt)
or in complex form:
(4.6)
c = 6e
Since viscosity produces a time lag in the stress-strain relationship, as revealed through
the derivative of the sinusoidal strain, the relationship becomes:
-= E1 Ce
+ E irie'
(4.7)
To simplify, let
r7 = tan(S)
(4.8)
such that
0-
=
E 6 sin(cot + 3)
(4.9)
where
E=E
1+2
(4.10)
20
Below is a graphical representation of the stress strain deformation relations.
Stress-Strain Relationship
1.5
1
0.5
0
-
Strain
----
Stress
-0.5
-1
-1.5
Time
Figure 4.1 Stress Strain Relationship
21
5.0 Materials
5.1 Asphalt
Asphalt is considered for use as a damping material because of its use in road pavement.
The loading of pavement structures is due to mobile forces, and the contact with the
surface unevenness of the pavement causes a transient state of stress. Systematic
experimental testing of real highway pavements and asphalt have been carried out, which
have correlated well with theoretical models of pavement structures and their dynamic
behavior.
Asphalt primarily consists of aggregate and binder materials. Aggregate sizes range from
3.35 mm to 60 microns, and may be natural crushed rock, gravel, or sand, or artificial, as
in slag or calcined bauxite. Today, tar has been replaced by bitumen for the binder
material. Bitumen is a hydrocarbon, coming off at the heavy end of the distillation of
crude oil.
The bituminous materials in asphalt have distinct features of viscoelastic behavior.
Below are selected data gathered from experiments conducted over several years by G.
Martincek and his team at the Institute of Civil Engineering and Architecture at the
Slovak Academy of Sciences, in Bratislava, Slovak Republic. Measurements were
performed using mechanical impedance, which is the complex ratio of the exciting force
to the velocity of motion at this point. The results of measurements at temperature T =
22
10, 20, and 400 C, in the form of isochrones and damping parameter variations are shown
below for a dense-cover asphalt layer. The values of the damping parameter Delta are
average values because it does not present a regular and distinct change with frequency.
However, the dependence of Delta on temperature T is very strong. In addition, the
complex modulus is very dependent on T and vibration frequency.
MPa3)
-OHz
7
2500
5002-0.
-
50000
000
0.400
22500
/
200
1000
10
+25
+40
T(C)
Figure 5.1 Isochrones for Typical Flexible System Asphalt
23
5.2 Viscoelastic Polymers
Natural rubber bearings have been used as damping devices in bridges for almost 40
years. Plastics are materials made from long molecules (Polymers). Generally Plastics
contain numerous chemical additives to refine their mechanical properties. Plastics are
widely used throughout industry because they can be formed into complicated shapes at a
very low cost using mass production methods. The range of properties available from
plastics materials has made them the prime choice for many applications. Plastics are
light, and durable. They generally are not able to withstand high temperatures and they
are not as strong as metals.
A rubber/elastomer is a polymeric material with long flexible molecular chains and the
ability to deform elastically when vulcanized. During the vulcanization process, rubber
molecules are linked with adjacent molecules at intervals along their lengths, usually by
sulfur to form a cross-linked elastic material that is stable over a wide range of
temperatures.
Another unique characteristic of rubber is that its modulus of elasticity is a complex
quantity, having both a real and imaginary component. Furthermore, this complex
modulus varies as a function of many parameters, most important of which are the
temperature and frequency of a given application. It is therefore necessary to establish an
accurate and thorough understanding of these parameters in order to design effective
damping treatments.
24
Below is the nomograph for a typical damping polymer, made by 3M.
242 Viscoelastic Damping Polymer Nomograph
Temperature (OC)
1000
-a-
10
0
0
8
10 0
SO
1000
...
Loss Facto____
-
.1.. --
F0g r.
-----
-------
.2.....graph
10
----------
.
-
--
3M..42
.r
0
.0-
_________g
P
LsFiguore 52 Noog...o.3.42DapngPlye
25
6.0 Mathematical Model
6.1 Single Degree of Freedom System
For simplicity, a building will be modeled as a single degree of freedom system subject
do ground motion, as shown below.
t
U + Ug
t
Ug
m
C
k
Figure 6.1 Schematic of Single Degree of Freedom System
The equation of motion is
mii(t)+ctd(t)+ k u(t) = -miig (t)
(6.1)
For design purposes, let the maximum relative motion of the mass to the ground be a
typical value of
U* =
.1m
(6.2)
26
Assume that the building, idealized as an SDOF system, behaves like a continuous-time
linear time-invariant filter described by the ordinary differential equation (6.1) and
generalized below
a
n
+a
dt"
d"_y
"n-
a
+......+
dt"~1
d 1y
dt 1
b
+ ay
d M +bi
'" dt'"
d'M-lu
+...+
dt'"'dt'i
diu
+b
1
(6. 3)
where y(t) denotes the system output subject to input u(t).
Since the Laplace Transform of the derivative of a function is given by
L df}
=
(6.4)
s" F(s)
then the system output in the frequency domain is given by
_s'"- + ....... +b s' +b
b s'" +b
Y(s) =
ans"
1
'"M1n-
'"
+ anl
s"
+.
0 U (S)
(6.6)
Y(s)=H(s) U(s)
H(s)=
bs'
mn
+b
(6.5)
as + ao
sm
n-1
ass + an s"
+.......+b s' +b
1
+.........+
(6.7)
as + ao
where H(s) is known as the transfer function for the system.
For the ODE specified in equation (6.1), where the input function is ug(t) and the output
function is u(t), the transfer function is
27
2
2
ms
+cs+k
H(s)=
(6.8)
Dividing the numerator and the denominator by m yields
2
H(s) =
2
2
(6.9)
k
Since the Laplace transform cannot be solved in closed form for the input motion ug(t),
the system equation must be put back into the time domain for processing the output. If
for the frequency domain, the output is found by equation (6.6), in the time domain the
output is found by
y(t) = h(t) ® u(t)
(6.10)
or the convolution integral
y(t) = Jh(v)u(t - r)dr
(6.11)
Therefore, the time domain impulse response for the transfer function must be found,
using the inverse Laplace transform.
1It is noted that the actual natural frequency of the system is w = wn*( 1-xi2)A( 1/2)
28
h(t)=
2;c j=
H (s)estds
1
2ff]j,
2s
ms 2 +cs+ k
eds
(6.12)
Once system variables such as damping ratio and natural frequency are specified, one can
solve for the inverse Laplace transform by the use of look-up tables, Matlab functions, or
partial fractions in closed form.
6.2 Damping Device
Assuming a design shown in Figure (9.2), where sheets of damping material could be
placed between the column and footing or underneath the base floor slab of a house, the
behavior could be modeled using the formulations in the previous section. The inertia
force due to gravity will be neglected in this study because the nature of the vibrations in
question (microtremors) do not cause significant displacements.
In deriving the stiffness, k, and the equivalent viscous damping coefficient, c, of the mat,
let
o-(t)L
u(t)=e(t)L= E*
E*
P(t)L
E*
E*A
(6.13)
29
such that
kEIA
1
k=
L
(6.14)
where L is the thickness of pad of damping material.
For equivalent viscous damping, the stress is proportional to the rate of strain as in
equation (4). An expansion of the above equations yields:
AE,
F = A u 9 [sin Ot +q cos Kt]
L
(6.15)
Such that the energy dissipated per cycle is
AE1 -2
Wviscoelastic =
__"7 __
-
f7L
(6.16)
g
Comparing this with the energy dissipation for a 100% viscous damper
d2
Wvi,,,,, = cn
(6.17)
yields
Cequivalent
-
C
AE 1 q7
-
(6.18)
This is reasonable because the viscosity of the material, and thus the damping effect,
should vary inversely with the rate of loading.
30
6.3 Preliminary Analysis: Pure Sinusoidal Excitation
For design purposes, let us assume that the excitation is deterministic and can be
expressed as a sinusoid or a sum of sinusoids, with zero phase for simplicity.
u g(t) =
a gsin(Qt)
(6.20)
The input motion in the frequency domain is merely the inverse Laplace transform of the
ground motion.
Ug(S) = L-1ug(t) = L-{
,
sin(Q t)
-
}
_
_
_
(6.21)
S2 +Q2
gsinQt)
Since it holds that the system output is equal to the system transfer function multiplied by
the input, in the frequency domain, the SDOF resultant relative motion response is
2
U(s)
= H(s)Ug (S) =
2
s +2o
Ug.
+ C
+
2
(6.22)
S2 + Q2
Solving for the response in the time domain requires applying the inverse Laplace
transform to this fourth order rational function. The solution yields:
u gsin(Qt - 3)
2
Q2 ) 2
Wn -n2)2
t= arctan-
-
+(2 ;Ojn
2;Ct nQ.
Cn2
_
2
2
Ii
(6.23)
(6.24)
aQ cos(Qt - 3)
d(t)=v(t)=
V(C02
Q 2 ) 2 +(2co i)2
(6.25)
31
7.0 Simulation Procedure
7.1 Model Parameters
Although it is very difficult to predict the exact frequency of micretremor excitation due
to traffic (as explained in section 6.2), for this analysis, we will assume a frequency
ranging from 5 Hz to 30 Hz for automobile and railway traffic. This is consistent with
observations made by the Swedish Council for Building Research and the Imperial
College of Science and Technology, London.
In addition, for ease of analysis, we will consider two cases for building type, which will
only affect the natural frequency for our simplified SDOF system. The first case is a
moderate size home, 2500-3000 sq-ft, with wood framing and possibly a few concrete
columns. The participating mass for this structure would be about 200,000 lbs. or
100,000 kg. The second case is a moderate size, short and stiff, brick or concrete office
building with a participating mass of 1,500,000 kg.
We will consider two materials for damping: asphalt and viscoelastic polymer. For
asphalt, Figure 5.1 reveals that the magnitude of the complex Elastic Modulus is between
1800 and 4000 MPa at 250 C for the loading frequency range 5 Hz to 30 Hz respectively,
with a loss factor of 0.7.
32
For viscoelastic polymers, values for compressive (normal) storage moduli could not be
obtained. However, for the frequency range and temperature of interest, this will be
approximated by the shear storage modulus and loss factor. For 3M Ultra-pure
Viscoelastic Damping Polymer (242F01/242F02/242F04), using Figure 5.2, the storage
modulus is 1 MPa and loss factor is 1 for 5 Hz and 1 MPa and 3 for 30 Hz. The
parameters for the test matrix are summarized below.
Material and Property
Frequency
Case 1: House
M = 100,000 kg
Case 2: Office Building
M = 1,500,000 kg
Asphalt: 5 Hz
On =
5, 40 Hz
(On =
5, 40 Hz
on =
5, 40 Hz
O, =
5, 40 Hz
on =
5, 40 Hz
=
5, 40 Hz
El = 1579 MPa, n = 0.7
Asphalt: 30 Hz
O
El = 3508 MPa, n = 0.7
Polymer: 5 Hz
E l ymer:a nz
El = I MPa, n = 1
Polymer: 30 Hz
CO
=
5, 40 Hz
=
5, 40 Hz
El = 1 MPa, n = 3
Table 7.1: Simulation Test Matrix
33
7.2 Controlling Velocity
The problem concerns controlling the building velocity to under the threshold levels for
human annoyance, since it is the more stringent requirement compared to building
damage levels.
Vt
mm
Substituting in mass values of 100,000 kg and 1.5 Mkg and excitation values of 5 Hz and
30 Hz, and max ground velocity in equation (6.25) and relating the material properties to
the natural frequency and damping ratio, the relationship between the normal Elastic
storage modulus and the loss factor can be expressed graphically to show the necessary
values to hold the peak total velocity to under 1 mm/s.
Below are graphs showing the relationship between the response velocity and the natural
frequency of the damped structure, considering excitations of 5 Hz and 30 Hz
(approximately 30 rad/s and 200 rad/s), with varying damping ratios. It is clear that if the
natural frequency of the structure in the vertical direction falls in this range, it is
susceptible to resonant phenomena and will need heavy damping to limit the relative
velocity response to below 1 mm/s. For the figures below, this translates to a ratio of
response magnitude to excitation magnitude, or amplification, of 1 [mm/s]/ 10 [mm/s],
which equals 0.01.
34
--
-7,--
--
EMPff-
x 10' Amplifaction of Relative Velocity for 30 Hz Vertical Excitation
1.2
-
Damping Ratio 1%
4%
-
10%
CL
1
(0
0)
0.8
0.6
CL
0.4
0.2
50
100 150 200 250 300 350 400 460 600
Natural Frequency of SDOF on Damping Pad in rad/s
550
Figure 7.1 Amplification of Velocity for Natural Frequency 30 Hz
Figure 7.2 Amplification of Velocity for Natural Frequency 5 Hz
35
I-,!Qi&
Actually, for the case of 30 Hz excitation, the amplification of relative velocity is so
small that the values already fall below threshold values. So the main concern is to keep
the natural frequency away from 5 Hz or 30 rad/s, or to make sure the material provides
sufficient damping. To find the optimum storage modulus, we make the following
calculations.
rad
20->
C,
> 40
S
S
> co2 , 10
$
4oofrd
40rad
40 0
rad
rad
2
00 rad)2
E
O2n1A
2
> 1600
>
4000000N > EA
house
> 16000000N
60000000N > E A
> 240000000N
where the height of the pads have been assumed to be 0.1 m. Considering the order of
magnitude for the necessary stiffness, it seems that polymer dampers would work best for
the case of a house and asphalt would work best for an office building. Using values for
the storage modulus of the individual materials, with respect to an excitation frequency
close to 5 Hz, the required areas are:
4m 2 > Alpoiymer
house
> 16m2
0.0375m 2 >
Aasphalt,office
> 0.15m 2
36
7.3 Input Vibration Data
Under the supervisory of the Central Artery and Tunnel Project of Boston,
Massachusetts, engineers from Bechtel/Parsons Brinckerhoff collected vibration data
from traffic passing over the elevated surface artery. These time histories will be used as
a starting point for the evaluation of damping materials to mitigate such traffic vibrations
from affecting nearby buildings.
The available data comes from instruments placed on grade, 15-20' away from a bent
column of the elevated highway, in the North End of Boston. The seismographs were
sensitive to 100 mV/g accelerations and were broad-band, collecting data from 5 Hz 20,000 Hz. The instruments sampled vibrations at 1 sample per second. Because the
data is not band-limited and the sampling frequency is well below the Nyquist sampling
rate, no frequency domain processing could be done because aliasing would otherwise
occur.
For four data sets taken on different days, the sample averages for the peak particle
vertical velocity and mean vertical velocity are 4.5 mm/s and 0.3 mns, respectively.
These are consistent with the values shown in Figure 2.3 for the variation of parameters
such as vehicle speed and soil type.
37
Because the samples were not taken often enough to accurately describe the
microtremors, these samples of the random variable, or noise, will be combined with
deterministic sinusoidal excitations of similar magnitude (10 mm/s), with frequencies
between 5 Hz and 30 Hz, encompassing the zone of possibility for vibrations due to
traffic. This is done, as opposed to merely applying a pure sinusoidal ground motion, in
order to study the effects of the random or nondeterministic characteristic of the
excitation.
A sample time history is shown below. The first being the original data set, and the
second being the interpolated data set (interpolated to a sufficient sampling rate), using
Matlab's INTERP, which resamples the data at a higher rate using a symmetric lowpass
interpolation, so that the mean square error between the points and their ideal values is
minimized.
Figure 7.3 Sample of Original Recorded Displacement Time History,
Sampled at 1 Hz
38
F
1
Resampled at 200 Hz
Input Displacement Time History
74rpolate
ne
4d
3
E2
-
-2
-3
0
0.5
1
15
2.5
2
Time [s]
3
3.5
4.5
4
X1
Figure 7.4 Interpolated Displacement Time History, Sampling Rate 200 Hz
7.4 Model and Test Procedure
SDOF models based on the transfer function expressed in equations (6.8) and (6.9) were
formed using the material properties shown in Table 7.1. They were loaded with vertical
excitations created by adding deterministic sinusoidal functions to the random data sets of
displacements collected from the Central Artery/Tunnel Project. The models based on
material properties assuming 5 Hz excitation frequency were loaded with a 5 Hz
frequency, and the same for 30 Hz. In addition, the models were also loaded with an
excitation that consisted of a sum of sinusoids with frequencies 5, 10, 20, and 30 Hz all
together. To study the total affect of the damping material, natural frequencies for the
39
SDOF systems were chosen at 5 Hz and 40 Hz; the former to check damping, latter to
check for isolation. The Matlab script used to run the simulation is included in the
Appendix.
40
8.0 Results
As a simplified, first pass approach, the simulation concluded that a polymer would
outperform asphalt for damping the microtremors studied here. This is not a surprise
because rubber has a lower stiffness and higher loss factor. However, in choosing a
damping device, there are other considerations, including ease of installation and cost.
Asphalt would obviously be much cheaper than a rubber bearing, and perhaps easier to
install for structures such as moderate sized homes, where the asphalt could be
sandwiched between the soil subgrade and the concrete foundation.
In addition, neither material could attenuate ground motion sufficiently to below
threshold values for human perception or annoyance across the board. However, for the
case where the thickness of the material was sufficient to result in a natural frequency
away from that of traffic vibration (5 - 30 Hz), levels were maintained at 1 mm/s. When
the excitation frequency was not driving at resonance, the materials were able to attenuate
the velocities enough to within recommended levels for Type I, II, and III buildings.
From this, we can also conclude that although damping ratios were very high, at
resonance, where damping was needed the most, the materials could not sufficiently
damp the structural vibrations. In the worst case, the velocity was amplified to six times
41
the input magnitude. This is not obvious when looking at the material properties and the
impulse response of the system. An example is shown below.
Impulse Response for Case 1: House3
0.2
0
T
E -0.2
C
0=
w-0.4
08
-0.8
-0.8
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
Time [s]
Figure 8.1 Sample of System Impulse Response
One of the problems is that in order for damping to be effective, the rate of displacement,
or velocity of the structure, must be large enough to counter the large inertia coming from
the mass of the structure. However, for microtremors, the displacements and velocities
are too small to effectively dissipate energy for this scale. This is discussed further in the
Hysteresis Loops section of the report. Below are the power spectral densities showing
the frequencies apparent in the resulting displacements and the levels. Because the
resulting frequencies are almost exactly at the natural frequencies, even when driven by a
multitude of frequencies, it is clear that damping was not achieved.
42
x 10"
1
Power Spectral Densities of Response for Case 1: House
-Asphalt
-
0.81
0.7
5 Hz
Asphalt 30 Hz
Polymer Hz
Polymer 30 Hz
-
0.9
-
0.6 -
0.5 -
0.4
-
03 -
0.2
-
0.1 01
0
5
30
25
20
15
Response Frequency [Hz]
0
0
35
Figure 8.2 Case for Structural Natural Frequency is 5 Hz,
Driving Frequencies: 5, 10, 20, 30 Hz
x 10'8
Power Spectral Densities of Response for Case 1: House
-
-
6
-
Asphalt 5 Hz
Asphalt 30 Hz
Polymer 5 Hz
30 Hz
.Polymer
5
2
0
5
10
20
15
Response Frequency [Hz
25
30
Figure 8.3 Case for Structural Natural Frequency is 5 Hz,
Driving Frequencies: 5, 10, 20, 30 Hz
43
Practically, it is difficult to achieve high levels of damping for these viscoelastic
materials responding to microtremors. The damping energy is converted to deformation
and heat, and the materials exhibiting highly viscous behavior tend to become more
viscous with increasing temperature and also stiffer with increasing number of cycles.
Therefore, the dampers lose effectiveness as the duration increases, unless a large volume
of material is used.
8.1 Hysteresis Loops
In the study of vibration, hysteresis loops as measures of energy dissipation are very
often applied. A hysteresis loop can be obtained by recording the magnitude of a force
versus the displacement brought about by its action. If there exists any damping in the
system then the graphical representation of this recording resembles a loop. The area
enclosed by the loop is proportional to the amount of energy dissipated in the system
within one period. There are different possible interpretations of hysteresis loops
according to the kind of force recorded during measurements.
a.) External loop - reflecting the relationship between externally applied loads and the
displacements measured in the system.
b.) Internal - if displacements are related to internal forces in the system.
c.) Damping - when damping forces are taken into consideration
44
The materials did exhibit damping, as shown by sample polymer hysteresis loops below.
However, as discussed earlier, it was not sufficient to attenuate the vibrations to below
threshold levels for human annoyance.
Hysteresis Loop. Case 1: House3, 5 Hz Excitation
0.3
0.2
0.1
E
M
-0.2
-0.3
-0.4-
-4
-3
-2
0
1
-1
Displacement [mm]
2
4
3
X 10 -
Figure 8.4 Sample of Hysteresis Loop for Polymer Bearing with 5 Hz Excitation
Figure 8.5 Sample of Hysteresis Loop for Polymer Bearing with 30 Hz Excitation
45
9.0 Feasibility
9.1 Vertical Load Capacity
For the application studied here, the compressive strength of a polymeric bearing depends
on the type of rubber, the bearing plan size, the bearing shape factor, and the thickness.
If elastomeric rubber is used, like that used for high damping rubber isolators, then the
compressive strength is sufficient and assisted by steel plates that prevent the rubber from
bulging, so that the bearing can support higher vertical static loads with only small
deformations.
For asphalts, the figures below show the relationship between the approximate thickness
and allowable bearing strength. For the thickness of 0.im assumed in this study, the
maximum vertical is well within the limits of asphalt.
46
-Design of the Flexible System
167
z1000
z
W
S800
LU
9600
z
STANDARD DEPTH SECTION OF 9-.
USED REGARDLESS IF SUBGRADI
BEARING TESTS MORE THAN 400 PS.I.
400
200
D)
6
.6
7
............
1......J *I. Im
8
9
I I I I - .I
I
10 .11
12 13 14 15 16 17 18 19 20 21
TOTAL PAVEMENT AND BASE THICKNESS
ft f
22 23 24
(INCHES)
FIG. 9-16.
Flexible-pavement design curve.
Figure 9.1 Compressive Strength of Typical Asphalt Layer
(Rogers and Wallace 1958)
9.2 Possible Design of Isolation System
To isolate the building from microtremors like those caused by vehicular or railway
traffic, pads could be inserted as resilient bearings on top of the pile caps or on the
footings. A sketch of such an installation is shown below. The bearings, about 100mm
thick when unloaded, are bonded with a self-levelling epoxy grout to the pile cap or to a
ground beam. A beam is placed on top to distribute the loads. A failsafe upstand can be
placed in between some of the pads to take the load in the event of a complete failure.
47
-q
Where a large load has to be supported on a single pile cap, a cluster of 10-30 rubber
pads could be used, with failsafe strips in between.
tawr Maflmg E
-
cover with sand in-fill
Rubber Pad with
Upstand
E1t1g3upiar
-
Culumn Sm FIa*
S-
-hmeded
.
Embed Pbiu
.a
Rodb
.L3
A;%
Arahw t1t EmienrW
- Anhar SO
mbenent
Figure 9.2 Design Schematic for Possible Column/Footing and
Damping Material Connection
The ratio of dynamic to static stiffness for natural rubber at these stress levels and at a
frequency of say 10Hz is between 1.7 and 2. The ratio for synthetic rubber is somewhat
higher. There is a marked effect of the shape factor on the natural frequency of a pad
design; pads with the same thickness and with a smaller plan area will normally have a
lower natural frequency for the same stress. The shape factor is defined as the ratio of
the loaded area to the area free to bulge. A better isolation can therefore be obtained by
installing a large cluster of small pads than only one or two very large ones. There is, of
course, a limit to this idea because the stability of the pads must be maintained. For a
48
typical damping material, TICO CV/CA, with thickness 25 mm, tests have indicated a
loss factor of 0.12. This corresponds to a damping ratio of 6%.
If there was any moment capacity considered in the column/footing connection in the
original design, then moment connections for the rubber or asphalt damper, would be
needed, as in most connections for rubber bearings used for earthquake shear baseisolation. This detail would be more easily applied in new construction; retrofits would
be more difficult. For a steel column connected to a pile cap or footing, the design would
mimic that for shear base-isolators. For concrete columns, the column would be spliced,
with the bearing placed in between. If it is a retrofit, the load would have to be
transferred away from the column using standard methods. For small structures, like
houses, which normally sit on a concrete slab, the bearings could be placed underneath
the slab, directly on to the soil foundation.
9.3 Other Materials: Foam
Expanded polystyrene (EPS), or Geofoam, is a very common product that is widely used
for packaging and in building construction. Geofoam is used as infill, as opposed to earth
materials which are heavy and can cause undesirable settlement or instability.
Manufacturing of EPS blocks begins with expandable polystyrene resin beads that are
49
generally less than 3 mm in diameter and contain microscopic cells filled with a blowing
agent. Because of its mechanical structure, Geofoam has proven to be an excellent
absorber of small ground vibrations. Unfortunately, it is not viable as a pad to place
underneath a structure because the comperessive strength is very low. At 1% strain, the
compressive resistance is only 75 kPa, which is not sufficient to take the load of a
supporting column.
50
Conclusion
From this study, it is clear that to attenuate ground born microtremors due to vehicle or
railway traffic, the ideal situation is to block the vibrations from getting into the building.
This can be done by surrounding the footings or piles by a concrete wall, or filling the
nearby area with Geofoam.
However, if this is not an option, a building could be retrofitted with polymeric bearings
to sufficiently attenuate vertical vibrations, as long as the bearings were large enough to
bring the apparent vertical natural frequency away from the 5 - 30 Hz resonant range.
This would in effect be more of an isolation approach, as opposed to damping.
Otherwise, the material cannot sufficiently dampen the structure. In addition, if cost is a
constraint, asphaltic dampers could be installed, that could sufficiently maintain levels to
resist structural damage, but not human annoyance.
51
Appendix A: Results from Simulation
A.1 StructuralNatural Frequency 5 Hz, Excitation Amplitude 10 mm/s at 5 Hz
Structure and
Material Property
House
Max Velocity
[mm/s]
0.0138
RMS Velocity
[mm/is]
0.0097
Min Velocity
[mm/s]
-.0300
0.0138
0.0097
-0.0300
0.0096
0.0068
-0.0293
0.0096
0.0068
-0.0293
Asphalt 5 Hz
Office
Asphalt 5 Hz
House
Polymer 5 Hz
Office
Polymer 5 Hz
A.2 Structural Natural Frequency 5 Hz, Excitation Amplitude 10 mm/s at 30 Hz
Structure and
Material Property
Max Velocity
[mm/s]
RMS Velocity
[mm/s]
Min Velocity
[mm/s]
House
0.0094
0.0066
-0.0752
0.0094
0.0066
-0.0752
0.0093
0.066
-0.0304
0.0093
0.066
-0.0304
Asphalt 30 Hz
Office
Asphalt 30 Hz
House
Polymer 30 Hz
Office
Polymer 30 Hz
52
A.3 Structural Natural Frequency 5 Hz, Excitation Amplitude 10 mm/s at 5, 10, 20,
and 30 Hz
Structure and
Material Property
House
Asphalt 5 Hz
Office
Max Velocity
[mm/s]
0.0224
RMS Velocity
[mm/s]
0.0147
Min Velocity
[mm/s]
-0.030
0.0224
0.0147
-0.030
0.0208
0.0125
-0.0293
0.0208
0.0126
-0.0293
0.0880
0.0581
-0.0752
0.0880
0.0581
-0.0752
0.0269
0.0177
-0.0304
0.0269
0.0177
-0.0304
Asphalt 5 Hz
House
Polymer 5 Hz
Office
Polymer 5 Hz
House
Asphalt 30 Hz
Office
Asphalt 30 Hz
House
Polymer 30 Hz
Office
Polymer 30 Hz
A.4 Structural Natural Frequency 40 Hz, Excitation Amplitude 10 mm/s at 5 Hz
Structure and
Material Property
House
Max Velocity
[mmn/s]
0.00026
RMS Velocity
[mm/s]
0.00008
Min Velocity
[mm/s]
-.0011
0.00026
0.00008
-.0011
0.00014
0.00007
-0.0008
0.00014
0.00007
-0.0008
Asphalt 5 Hz
Office
Asphalt 5 Hz
House
Polymer 5 Hz
Office
Polymer 5 Hz
53
A.5 Structural Natural Frequency 40 Hz, Excitation Amplitude 10 mm/s at 30 Hz
Structure and
Material Property
House
Asphalt 30 Hz
Office
Asphalt 30 Hz
House
Polymer 30 Hz
Office
Polymer 30 Hz
Max Velocity
[mm/s]
0.0055
RMS Velocity
[mm/s]
0.0039
Min Velocity
[mnims]
-0.0056
0.0055
0.0039
-0.0056
0.0015
0.0011
-0.0015
0.0015
0.0011
-0.0015
A.6 Structural Natural Frequency 40 Hz, Excitation Amplitude 10 mm/s at 5, 10, 20,
and 30 Hz
Structure and
Material Property
House
Max Velocity
[mm/s]
0.0019
RMS Velocity
[mm/s]
0.0007
Min Velocity
[mm/s]
-0.0039
0.0019
0.0007
-0.0039
0.0013
0.0005
-0.0028
0.0013
0.0005
-0.0028
0.0116
0.0028
-0.0114
0.0116
0.0028
-0.0114
0.0028
0.0009
-0.0052
0.0028
0.0009
-0.0052
Asphalt 5 Hz
Office
Asphalt 5 Hz
House
Polymer 5 Hz
Office
Polymer 5 Hz
House
Asphalt 30 Hz
Office
Asphalt 30 Hz
House
Polymer 30 Hz
Office
Polymer 30 Hz
54
Appendix B: Matlab script used to run simulation
3
%%
%%
%%
%%
%%
%%
%%
%%
Diane Floresca
Massachusetts Institute of Technology
Department of Civil and Environmental Engineering
Masters of Engineering Thesis 2003
Research Into Building Vibrations
Using Aphalt or Polymers to Dampen Traffic Vibrations in
Buildings
%%
%%
%%
%%
%%
The following script was written to simulate 2 SDOF systems
which model a typical house structure and a small office
building, with asphalt and polymer pads placed between the
building and the foundation.
%%
%%
%%
%%
E
=
%%
%%
%%
%%
[1579000000 3508000000 1000000 1000000];
L = 0.1; %100 mm
n = [0.7 0.7 1 3];
m = [100000 1500000];
omega = 2*pi*[5 30 5 30];
%Set target vertical period to specified frequency of study
%Frequencies studied: 31, 50, and 250 rad/s
A_house = (E.^(-1)).*((3 A2)*m(l)*L);
A_office = (E.^(-1)).*((3 A2)*m(2)*L);
for i=1:4
kjhouse(i) = E(i)*A house(i)/L;
= A-house (i) *E (i) *n (i)/ (L*omega (i));
c-house (i)
wnhouse(i) = (k-house(i)/m(1))^(0.5);
xihouse(i) = c-house(i)/(2*wn-house(i)*m(l));
k_office(i) = E(i)*A-office(i)./L;
c_office(i) = A-office(i)*E(i)*n(i)/(L*omega(i));
= (k-office(i)/m(2))^(0.5);
wnoffice(i)
xioffice(i) = coffice(i)/(2*wnoffice(i)*m(2));
end
%Graphing the theoretical amplification
w = 0:2:200*2*pi;
xi = [0.01 0.04 0.1];
factor for velocity response
55
vamp-housel-xl =
((((w.^2)(omega(l)^2)).^2)+((w.*2*xi(1)*omega(1)).^2)).^(-1/2);
vamp-houselx4 = ((((w.^2)(omega(1)^2)).^2)+((w.*2*xi(2)*omega(1)).^2)).^(-1/2);
vamp_housel-x10 = ((((w.^2)(omega(1)^2)).^2)+((w.*2*xi(3)*omega(1)).^2)).^(-1/2);
figure
plot(w,vamphousel_xl,w,vamphouselx4,w,vamphouselxlO);
xlabel('Natural Frequency of SDOF on Damping Pad in rad/s');
ylabel('Amplification of Relative Velocity Response');
title('Amplifaction of Relative Velocity for 5 Hz Vertical
Excitation');
legend('Damping Ratio 1%','4%','10%');
vamp-house2_x1 = ((((w.^2)(omega(2)^2)).^2)+((w.*2*xi(1)*omega(2)).^2)).^(-1/2);
vamp-house2_x4 = ((((w.^2)2
(omega(2)^2)).^2)+((w.*2*xi(2)*omega(2)).^2)).^(-1/ );
vampjhouse2_x1O = ((((w.^2)(omega(2)^2)).^2)+((w.*2*xi(3)*omega(2)).^2)).^(-1/2);
figure
plot(w,vamphouse2_x,w,vamphouse2_x4,w,vamphouse2_xl0);
xlabel('Natural Frequency of SDOF on Damping Pad in rad/s');
ylabel('Amplification of Relative Velocity Response');
title('Amplifaction of Relative Velocity for 30 Hz Vertical
Excitation');
legend('Damping Ratio 1%','4%','10%');
plot(w,vamphousel_x1,w,vamphouselx4,w,vamphousel-xlO,w,vamp-house2 _
xl,w,vamp-house2_x4,w,vamphouse2_xlO);
xlabel('Natural Frequency of SDOF on Damping Pad in rad/s');
ylabel('Amplification of Relative Velocity Response');
title('Amplifaction of Relative Velocity for 5 Hz and 30 Hz Vertical
Excitation');
legend('Damping Ratio 1% - 5 Hz','4% - 5 Hz','10% - 5 Hz','Damping
Ratio 1%
-
30 Hz','4% -
30 Hz','10%
-
30 Hz');
load d:\vibetestl.txt
%sampled at 1 Hz (uips)
signalvelocity =
((l0.A(vibetestl./10)) .A(1/2))*0.0254*0.000001;
%[m/si
%zero out the velocity
signalvelocity = detrend(signalvelocity);
%just taking 2000 points from input, somewhere in the middle
signaldisp = indefintegral(signalvelocity(1999:3999));
%Check statistics
rmsvelocity =
(trapz(signalvelocity.^2)/(length(signalivelocity)))^(0.5);
rmsdisp = (trapz(signal-disp. 2)/(length(signal_disp)))^(0.5);
overthreshold=length(find(signalbvelocity>0.001));
56
meani = mean(signalvelocity);
stdl = std(signal-velocity);
maxi = max(abs(signalvelocity));
rmsl= (trapz(signalvelocity.^2)/length(signal-velocity))^(1/2);
%Assuming highest frequency content to be 80 Hz or 160pi rad/s
%Nyquist = 160 Hz -- > 200 Hz
ws = 2*pi*200;
fs = 200;
ts = 1/200;
%Since data sampled at 1 Hz, we're going to insert points to make fs
200 Hz
%we will assume at this point that the input is band limited to 1 Hz
inputi = interp(signal_disp,200);
=
1=400000;
tl = 0:0.005:length(signal-disp)-l;
t1 = t1(1:1);
input5 = 0.0003*(inputl(1:1) + sin(2*pi*5*tl(1:1)));
input30 = 0.00005*(inputl(1:1) + sin(2*pi*30*tl(1:1)));
inputall = 0.0003*inputl(1:1) + 0.0003*sin(2*pi*5*tl(1:1))
0.00015*sin(2*pi*10*tl(1:1))+
0.00007*sin(2*pi*20*tl(1:1))+0.00003*sin(2*pi*30*tl(1:1));
+
%Creating Object Transfer Functions
s = tf('s');
H_housel
H_house2
H_house3
H_house4
H_officel
H_office2
H_office3
H_office4
=
=
=
=
(-m(1)*(sA2))/((m(l)*s^2)+(cJhouse(l)*s)+k-house(l));
(-m(l)*(s^2))/((m(l)*s^2)+(c-house(2)*s)+k-house(2));
(-m(1)*(sA2))/((m(1)*sA2)+(c-house(3)*s)+khouse(3));
(-m(1)*(sA2))/((m(1)*sA2)+(c9house(4)*s)+k-house(4));
=
=
=
=
(-m(2)*(s^2))/((m(2)*s^2)+(c-office(l)*s)+k-office(l));
(-m(2)*(sA2))/((m(2)*s2)+(c-office(2)*s)+k-office(2));
(-m(2)*(sA2))/((m(2)*s2)+(c-office(3)*s)+k-office(3));
(-m(2)*(sA2))/((m(2)*s2)+(c-office(4)*s)+k-office(4));
%Checking impulse response
impulseinput = 1+zeros(1,1000);
timpulse = 0:0.005:5;
timpulse = timpulse(1:1000);
[Yjhouse3_5_impulse,timpulsel = lsim(H-house3,impulseinput,timpulse);
title('Impulse Response for Case 1: House3');
xlabel('Time [s]');
ylabel('Response [mm]');
[Y-officel_5_impulse,timpulse] = lsim(Hofficel,impulseinput,timpulse);
title('Impulse Response for Case 2: Officel');
xlabel('Time
[s]');
ylabel('Response [mm]');
%Creating Response Time Histories for Inputs of Various Frequencies
[Y-housel_5,t]=lsim(Hhousel,input5,tl);
[Y-house3_5,t]=lsim(H-house3,input5,t1);
[Y-house2_30,t]=lsim(H-house2,input30,tl);
57
[Y-house4_30,t]=lsim(Hhouse4,input30,tl);
[Yofficel_5,t]=lsim(H_officel,input5,tl);
[Y-office3_5,t]=lsim(Hoffice3,input5,tl);
[Y-office2_30,t]=lsim(H-office2,input30,tl);
[Yoffice4_30,t]=lsim(H-office4,input30,tI);
[Yjhouselall,t]=lsim(H-housel,inputall,tl);
[Y-house3_all,t]=lsim(Hhouse3,inputall,t1);
[Y-house2_all,t]=lsim(Hhouse2,inputall,tl);
[Y-house4_all,t]=lsim(H-house4,inputall,tl);
[Y-officelall,t]=lsim(H_officel,inputall,tl);
[Y-office3_all,t]=lsim(Hoffice3,inputall,tl);
[Y-office2_all,t]=lsim(H-office2,inputall,tl);
[Y-office4_all,t]=lsim(Hoffice4,inputall,tl);
[V-housel_5] = diff(Y-housel_5)/ts;
[Vjhouse3_5] = diff(Y_house3_5)/ts;
[V-of f icel_5] = dif f (Yoff icel_5) /ts;
[V-of f ice3_5] = dif f (Y-off ice3_5) /ts;
[V-house2_30 ] = dif f (Yjhouse2_30) /ts;
[V-house4_30 ] = dif f (Yjhouse4_30) /ts;
[V-office2_30] = diff(Y-office2_30)/ts;
[Voffice4_30] = diff(Yoffice4_30)/ts;
[Vjhouselall]
[V-house2_all]
[Vhouse3_all]
[Vjhouse4_all]
[V-of ficelall]
[V-of f ice2_all]
[Vof f ice3_all]
[Vof f ice4_all]
=
=
=
=
diff(Yjhouselall)/ts;
diff(Yhouse2_all)/ts;
diff(Y-house3_all)/ts;
diff(Y-house4all) /ts;
/ts;
= dif f (Y-officel_all)
= dif f (Y-off ice2_all) /ts;
= dif f (Y-off ice3_all) /ts;
= dif f (Y-office4_all) /ts;
rmsVhousel_5 = rms(V-housel_5)
rmsVhouse2_30 = rms(V-house2_30)
rmsVhouse3_5 = rms(Vhouse3_5)
rmsVhouse4_30 = rms(V~house4_30)
rmsVofficel_5 = rms(Vofficel_5)
rmsVoffice2_30 = rms(V-office2_3O)
rmsV_of fice3_5 = rms(V--office3_5)
rmsVoffice4_30 = rms(V-office4_30)
rmsVhouselall
rmsVhouse2_all
rmsVhouse3_all
rmsVhouse4_all
rmsV_of ficel_all
rmsV off ice2_all
rmsVoffice3_all
rmsV_office4_all
=
=
=
=
rms (V-houselall)
rms (VXhouse2_all)
rms(Vjhouse3_all)
rms(V-house4_all)
=
=
=
=
rms(V-officel_all)
rms(V-office2_all)
rms(V-office3_all)
rms(Voffice4_all)
maxVhousel_5 = max(V-housel_5)
maxVhouse2_30 = max(V-house2_30)
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maxVhouse3_5 = max(V-house3_5)
maxVhouse4_30 = max(Vhouse4_30)
maxVofficel_5 = max(V-office1_5)
maxVoffice2_30 = max(VWoffice2_30)
maxVoffice3_5 = max(Voffice3_5)
maxVoffice4_30 = max(V-office4_30)
maxVhouselall
maxVhouse2_all
maxVhouse3_all
maxVhouse4_all
=
=
=
=
max(V-houselall)
max(V-house2_all)
max(V-house3_all)
max(V-house4_all)
=
=
=
=
maxVofficelall
maxVoffice2_all
maxvoffice3_all
maxVoffice4_all
max(VXofficelall)
max(V-office2_all)
max(Voffice3_all)
max(V-office4_all)
minVhousel_5 = min(V-housel_5)
minVhouse2_30 = min(Vhouse2_3O)
minVhouse3_5 = min(Vhouse3_5)
minVhouse4_30 = min(V\house4_30)
minVofficel_5 = min(V-officel_5)
minVoffice2_30 = min(V-office2_30)
minVoffice3_5 = min(Voffice3_5)
minVoffice4_30 = min(VXoffice4_30)
minVhouselall
minVhouse2_all
minVhouse3_all
minVhouse4_all
minVofficelall
minVoffice2_all
minVoffice3_all
minVoffice4_all
=
=
=
=
min(Vjhouselall)
min(Vjhouse2_all)
min(V-house3_all)
min(Vjhouse4_all)
=
=
=
=
min(V\officelall)
min(V-office2_all)
min(V-office3_all)
min(V-office4_all)
[p-houselall,f-housel-all]=psd(Y-house_all,1026,200);
[pjhouse2_all,fhouse2_all]=psd(Y-house2_all,1026,200);
[pjhouse3_all,f-house3_all]=psd(Yhouse3_all,1026,200);
[pjhouse4_all,f house4_all]=psd(Yhouse4_all,1026,200);
[p-officel_all,f-officel_all]=psd(Y_office1_all,1026,200);
[p-office2_all,f-office2_all]=psd(Y_office2_all,1026,200);
[p-office3_all,f-office3_all]=psd(Y_office3_all,1026,200);
[p-office4_all,f-office4_all]=psd(Y_office4_all,1026,200);
figure
plot(fhouselall,phouselall,f_house2_all,phouse2_all,f_house3_all,p
_house3_all,f-house4_all,p_house4_all);
title('Power Spectral Densities of Response for Case 1: House');
xlabel('Response Frequency [Hz]');
legend('Asphalt 5 Hz', 'Asphalt 30 Hz', 'Polymer 5 Hz', 'Polymer 30
Hz');
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figure
plot(fofficelall,p-officelall,f_office2_all,poffice2_all,f_office3_
all,poffice3_all,foffice4_all,poffice4_all);
title('Power Spectral Densities of Response for Case 1: office');
xlabel('Response Frequency [Hz]');
legend('Asphalt 5 Hz', 'Asphalt 30 Hz', 'Polymer 5 Hz', 'Polymer 30
Hz');
figure
plot(0:l:length(signalvelocity)-l,signal-velocity);
title('Original Input Velocity Time History -- Sampled At 1 Hz');
xlabel('Time [s]');
ylabel('Displacement [Imm/s]');
figure
plot(0:1:length(signal_disp)-l,signal_disp);
title('Original Input Displacement Time History -xlabel('Time [s]');
ylabel('Displacement [mm]');
figure
plot(0:1:length(inputl)-l,inputl);
title('Interpolated Input Displacement Time History
Sampled At 1 Hz');
--
Resampled at 200
Hz');
xlabel('Time [s]');
%ylabel('Displacement [mm]');
%Hysteresis Loops
h_house = xihouse.*wnhouse;
Fdhousel_5 =
Fdhouse3_5 =
Fdhouse2_30 =
Fdhouse4_30 =
-2*h-house(l)*Vhousel_5;
-2*hhouse(3)*Vhouse3_5;
-2*h-house(2)*V-house2_30;
-2*h-house(4)*V-house4_30;
h_office = xioffice.*wnoffice;
Fdofficel_5 = -2*h_office(l)*V\officel_5;
Fdoffice3_5 = -2*h_office(3)*VWoffice3_5;
Fdoffice2_30 = -2*hoffice(2)*V-office2_30;
Fdoffice4_30 = -2*hoffice(4)*V-office4_30;
Fdhouselall = -2*hhouse(l)*V-housel_all;
Fdhouse2_all = -2*hhouse(2)*VWhouse2_all;
Fdhouse3_all = -2*hhouse(3)*VXhouse3_all;
Fd_house4_all = -2*hhouse(4)*Vjhouse4_all;
Fdofficelall = -2*hoffice(l)*V\officel_all;
Fd_office2_all = -2*hoffice(2)*VWoffice2_all;
Fd_office3_all = -2*hoffice(3)*VXoffice3_all;
Fd_office4_all = -2*hoffice(4)*VXoffice4_all;
figure
plot(Y-housel_5(1000:5000),Fdhousel_5(1000:5000));
title('Hysteresis Loop, Case 1: Housel, 5 Hz Excitation');
xlabel('Displacement [mm]');
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ylabel('Damping Force [N]');
figure
plot(Y-house3_5(1000:5000),Fdhouse3_5(1000:5000));
title('Hysteresis Loop, Case 1: House3, 5 Hz Excitation');
xlabel('Displacement [mm]');
ylabel('Damping Force [N]');
figure
plot(Y house2_3O(lOOO:5000),Fdhouse2_30(1000:5000));
title('Hysteresis Loop, Case 1: House2, 30 Hz Excitation');
xlabel('Displacement [mm]');
ylabel('Damping Force [N]');
figure
plot(Yhouse4_30(1000:5000),Fdhouse4_30(1000:5000));
title('Hysteresis Loop, Case 1: House4, 30 Hz Excitation');
xlabel('Displacement [mm]');
ylabel('Damping Force [N]');
figure
plot(Yofficel_5(1000:5000),Fd-officel_5(1000:5000));
title('Hysteresis Loop, Case 2: officel, 5 Hz Excitation');
xlabel('Displacement [mm]');
ylabel('Damping Force [N]');
figure
plot(Y-office3_5(1000:5000),Fd-office3_5(1000:5000));
title('Hysteresis Loop, Case 2: office3, 5 Hz Excitation');
xlabel ( 'Displacement [mm]');
ylabel('Damping Force [N]');
figure
plot(Y-office2_3O(l000:5000),Fd-office2_30(1000:5000));
title('Hysteresis Loop, Case 2:
xlabel ( 'Displacement [mm]');
ylabel('Damping Force [N]');
figure
office2, 30 Hz Excitation');
plot(Y-office4_3O(lOOO:5000),Fd-office4_30(1000:5000));
title('Hysteresis Loop, Case 2: office4, 30 Hz Excitation');
xlabel('Displacement [mm]');
ylabel('Damping Force [N]');
figure
plot(Y housel-all(1000:399999),Fd-houseall(1000:399999));
title('Hysteresis Loop, Case 1: Housel, 5,10,20,30 Hz Excitation');
xlabel('Displacement [mm]');
ylabel('Damping Force [N]');
figure
plot(Y-house2_all(1000:399999),Fdhouse2_all(1000:399999));
title('Hysteresis Loop, Case 1: House2, 5,10,20,30 Hz Excitation');
xlabel('Displacement [mm]');
ylabel('Damping Force [N]');
figure
plot(Y house3_all(l000:399999),Fdhouse3_all(1000:399999));
title('Hysteresis Loop, Case 1: House3, 5,10,20,30 Hz Excitation');
xlabel('Displacement [mm]');
ylabel('Damping Force [N]');
figure
61
plot(Yhouse4_all(10OO:399999),Fdhouse4_all(1000:399999));
title('Hysteresis Loop, Case 1: House4, 5,10,20,30 Hz Excitation');
xlabel('Displacement [mm]');
ylabel('Damping Force [N]');
figure
plot(Y-officel all(1000:399999),Fd-officel_all(1000:399999));
title('Hysteresis Loop, Case 2: officel, 5,10,20,30 Hz Excitation');
xlabel('Displacement [mm]');
ylabel('Damping Force [N]');
figure
plot(Yoffice2_all(1000:399999),Fd-office2_all(1000:399999));
title('Hysteresis Loop, Case 2: office2, 5,10,20,30 Hz Excitation');
xlabel('Displacement [mm]');
ylabel('Damping Force [N]');
figure
9
plot(Yoffice3_all(1000:399999),Fd-office3_all(1000:39 999));
title('Hysteresis Loop, Case 2: office3, 5,10,20,30 Hz Excitation');
xlabel('Displacement [mm]');
ylabel('Damping Force [N]');
figure
plot(Yoffice4 all(1000:399999),Fd office4_all(lOOO:399999));
title('Hysteresis Loop, Case 2: office4, 5,10,20,30 Hz Excitation');
xlabel('Displacement [mm]');
ylabel('Damping Force [N]');
62
References
Bhowmick, Anil K. and Stephens, Howard L. Ed. Handbook of Elastomers. New York,
2001: Marcel, Dekker, Inc.
Connor, Jerome J. Introduction to Structural Motion Control. New Jersery, 2003: Prentice
Hall
Hollaway, L.C. Ed. Polymers and Polymer Composites in Construction. London, 1990:
Thomas Telford Ltd.
Holmberg, Roger. Vibrations Generated by Traffic and Building Construction Activities.
Sweden: 1984: Spangbergs Tryckerier AB.
Martin, J. Rogers and Wallace, Hugh A. Design and Construction of Asphalt Pavements.
New York, 1958: McGraw-Hill Book Company.
Martincek, G. Dynamics of Pavement Structures. Slovak Republic, 1994: E & FN Spon.
Nicholls, J.C. Ed.Asphalt Surfacings. New York, 1998: E & FN Spon.
Oppenheim, Alan V., Schafer, Ronald W. and Buck, John R. Discrete-Time Signal
Processing. New Jersey, 1998: Prentice Hall.
www.3m.com
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